Singular sets of smooth maps \(f:M \rightarrow F\)  of smooth \(n\) -manifolds into surfaces played a strong role in recent various discoveries. Studying singular sets of maps, Gay and Kirby [1]} proved that any smooth closed oriented connected 4-manifold admits a trisecting map to \(\mathbb {R}^2\)  in analogy to the existence of Heegaard splittings for oriented connected closed 3-manifolds, see also the paper [2]} by Baykur and Saeki for the existence of a simplified trisection. Kalmar and Stipsicz [3]} obtained upper bounds on the complexity of the singular set of maps from 3-manifolds to the plane. These upper bounds are expressed in terms of certain properties of the link \(L \subset S^3\) , where the 3-manifold is obtained via integral surgery along \(L\) . Ryabichev [4]} gave precise conditions for the existence of maps of surfaces with prescribed loci of singularities.
Yamamoto [5]} classified the existence of smooth maps between oriented surfaces with prescribed singularities by examining the number of components of the singular set. Kitazawa  [6]} studied simple fold maps (of non-negative dimension) of smooth manifolds to Euclidean target spaces (\(\mathbb {R}^2\) , in particular) whose singular sets are concentric spheres. Saeki [7]} and  [8]} showed that every closed connected oriented 3-manifold admits a stable map to a sphere without definite fold points.
In  [8]}, Saeki posed the following question.
